Problem: Simplify and expand the following expression: $ \dfrac{3p}{3p - 9}-\dfrac{p}{5p - 2} $
In order to subtract expressions, they must have a common denominator. Get both fractions over a common denominator of $(3p - 9)(5p - 2)$ Multiply the first term by $\dfrac{5p - 2}{5p - 2}$ $ \begin{align*} \dfrac{3p}{3p - 9} \times \dfrac{5p - 2}{5p - 2} & = \dfrac{(3p)(5p - 2)}{(3p - 9)(5p - 2)} \\ & = \dfrac{15p^2 - 6p}{(3p - 9)(5p - 2)}\end{align*} $ Multiply the second term by $\dfrac{3p - 9}{3p - 9}$ $ \begin{align*} \dfrac{p}{5p - 2} \times \dfrac{3p - 9}{3p - 9} & = \dfrac{(p)(3p - 9)}{(5p - 2)(3p - 9)} \\ & = \dfrac{3p^2 - 9p}{(5p - 2)(3p - 9)}\end{align*} $ Now we have: $ = \dfrac{15p^2 - 6p}{(3p - 9)(5p - 2)} - \dfrac{3p^2 - 9p}{(5p - 2)(3p - 9)} $ Now both terms have a common denominator we can subtract the numerators: $ = \dfrac{15p^2 - 6p - (3p^2 - 9p)}{(3p - 9)(5p - 2)} $ $ = \dfrac{15p^2 - 6p - 3p^2 + 9p}{(3p - 9)(5p - 2)} $ $ = \dfrac{12p^2 + 3p}{(3p - 9)(5p - 2)}$ Expand the denominator: $ = \dfrac{12p^2 + 3p}{15p^2 - 51p + 18}$ Simplify: $ = \dfrac{4p^2 + p}{5p^2 - 17p + 6}$